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Fundamentals of Probability Theory M FUNDAMENTALS OF PROBABILITY THEORY M. E. Harr, Purdue University The components of a pavement system, its loadings and responses, its con­ stitutive materials, and conditions of weather vary in time and location in a random manner. Mathematical models of such systems are known as stochastic processes. This paper presents some fundamentals of proba­ bility theory that form the building blocks of such processes. Specific topics treated are deterministic and stochastic systems, randomness and probability, tree diagrams, permutations and computations, conditional probabilities, independence, and Bayes' theorem. Examples are presented to demonstrate the use of the concepts relative to factors entering the analysis, design, construction, and proofing of pavement systems. The concept of chance as it applies to dice or cards is discussed. In this paper a collectfon of tools is described, and their use is demonstrated. •ACHIEVEMENTS in transportation technology during the last 2 decades have increased the need for pavement evaluation procedures with which to assess the future trends of pavement behavior. The rates and magnitudes of loadings imposed on today's pavement systems surpass those previously experienced, especially those due to air transport vehicles. The nature of these loadings places greater demands on pavements than they were designed and constructed for. The deterioration of today's pavements has become a major problem of civil engineering. The problem facing the profession today is not how to design and build new pavement systems for greater frequency and magnitude of loadings, but how to upgrade and pro­ vide the remedial measures for existing pavement systems to meet current and future traffic demands. In pavement design and analysis, factors that are commonly swept under the carpet in other analytical problem areas cannot be assumed away. For example, a pavement consists of distinct layers with unknown contacts at their interface; the layers may or may not be in contact in space or in time. Imposed loadings (wheels) are relatively large in area compared with the thickness of the surface layer; consequently, Saint­ Venant's principle cannot be invoked to change the system to an equivalent homogeneous and isotropic body. Ambient conditions greatly alter the properties of the layers, which range from thermal plastic, temperature-sensitive materials to granular soils whose actions depend greatly on their voids. Each layer is composed of complex conglomera - tions of discrete particles of varying shapes, sizes, and orientations. In addition, loads are variable in both magnitude and time and are dynamic in nature. It is not surprising how poor predictions of the transmission of induced energy through such systems have been. Randomness alone dictates the probabilistic (casual) rather than deterministic (causal) treatment. DETERMINISTIC AND STOCHASTIC SYSTEMS Systems that can be described by unique explicit mathematical relationships are said to be deterministic. An example of a common deterministic system is shown in Figure la. The system is composed of the mass m, suspended from a linear spring (with spring constant k) , which is displaced an amount l:J,. from its equilibrium position. The mathematical relationship of the response y(t) is 1 2 y(t) =~cos~ t ( 1) for t ;;, 0. This expression provides the unique position of the mass y(t) at any instant of time; hence, the system is completely determined or deterministic. A pictorial rep­ resentation of the model is shown in Figure lb. An example of the use of this model with respect to pavements was given by Harr (3). The concept of a stochastic system is shown in Figure 2, which illustrates a vertical cross section through a pavement subjected at its surface (the x-axis) to a unit force (say per unit length normal to the plane of the paper) acting at point x = x1 • Suppose that we wish to determine the magnitudes of the two forces FA and Fa, located as shown at equal distances a on either side of the unit force at a constant depth z = z 1• In effect, we would seek the transmission of the unit force through the pavement. A pavement, in its general form, is composed of a complex conglomeration of discrete particles, in arrays of varying shapes, sizes, and orientations, and contains randomly distributed concentrations of cementing agents. Certainly, we cannot expect that, in general, the forces registered by FA and Fa will always be equal. (In the deterministic approach, it is customary to plead symmetry and hence the equality of the two forces, i.e., FA = Fa.) In fact, because of the variations in the characteristics of such media, we should expect that they will seldom be equal because the location x = x1 varies. For example, if one of the forces was not in contact with a solid particle, i.e., was in a void, no force would be noted. Evidently, as the unit force is moved in time, as for moving vehicles, through a series of points x = x1, the magnitudes of the forces would be expected to be random in character, i.e., casual rather than causal. Systems that display random re­ sults with time are said to be stochastic. The thesis here is that, to be meaningful, ex­ periments involving such systems should be formulated in terms of probabilistic state­ ments rather than explicit expressions. RANDOMNESS AND PROBABILITY As noted, in the deterministic approach the outcomes of experiments (observations or phenomena) are treated as absolute quantities. For example, when given the total weight W, total volume V, and weights of solids W, for a water-saturated soil mass, the porosity can be obtained from the formula n = (W - W,)/Vy,.. In particular, given 3 W = 100 g, w. = 55 g, V = 100 cm , the porosity is n = 45 percent. Implied in this re­ sult is that, if we were to carry out the weighings and volumetric determinations on a large number of samples of the soil, under certain similar conditions, we would ex­ pect on the average that the ratio of volume of voids to total volume would be 0.45. Ob­ viously, the determi ned porosity would not be 45 percent in every experiment. Some­ times it would be 40 or 41 percent, other times 43 or 46 percent. Occasionally, it may even be very much smaller or very much greater than 45 percent. This example illustrates what is meant by random experiments, experiments that can give varying outcomes (results) depending on chance circumstances that are either unknown or beyond control. The distributions of particle sizes in a number of soil samples from the same test pit will not be the same. The variation in measured pave­ ment thickness for a given section will show considerable differences. Contrary to common belief, the inability to obtain concise descriptions of events or observations is not a declaration of ignorance; it is the way nature and the real world behave-fraught with uncertainty. Stated more succinctly, there is no absolute knowledge. What phys­ icists considered exact and ordered prior to the development of quantum mP.r.hanir.s turned out to be merely the mean value of a much more impressive structure. The diversity of results of apparently similar circumstances is the consequence of randomness. A single data set represents only one of many possible results that may occur. Each of these may be considered as a single result of a random experiment (or phenomenon), the collection of which produces a random process. In other words, a data record for a particular sample of a random phenomenon is only one physical real- 3 ization of a random process. Every test and measurement conducted on pavement systems introduces a magnitude whose numerical value depends on random factors that are beyond control. Further­ more, each such magnitude can have a different value in successive trials. This type of magnitude is called a random variable; the separate magnitudes are called elemen­ tary events. The outcomes of a random experiment are called elementary events if (a) only one outcome can occur at a time and (b) one outcome always does occur. Con­ dition a specifies that the outcomes are mutually exclusive or are disjoint; that is, no two elementary events can occur simultaneously. Condition b states that an elementary event is possible. The classic example of an elementary event is the outcome of the toss of a fair die. (Historically, questions relative to dice, asked of Pascal, precipi­ tated the mathematical theory of probability.) Condition a is satisfied because only one face can appear per toss of a die. Condition b is ensured because any one of the six faces is likely to appear. Implied in the die toss experiment is that the numbers 1, 2, 3, 4, 5, and 6 each have a possibility of occurring with equal likelihood. However, the number that will appear on any one toss is uncertain. Suppose now that the experiment is repeated many times. Even though the numbers shown on the faces may be different in successive tosses, it is reasonable to expect that, over the long run, any one number will occur one-sixth of the time. A gambler would say that the odds against tossing any specified number is five to one. The probabilist would define that probability to be one-sixth. The measure of the probability of an outcome is its relative frequency. That is, if an outcome E can occur n times in N equally likely trials, the probability of the occur­ rence of outcome E (after a large, theoretically infinitely large, number of experi­ ments) is n P(E) N (2) Also, implied in equation 2 is the concept of the ratio of favorable outcome to the num­ ber of all possible cases. This definition was first formulated by Laplace in 1812.
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